An iron cube of side 10 cm is hammered into a rectangular sheet of thickness 0.5 cm. If the sides of the sheet are in the ratio 1 : 5, the sides are

### Concept & Conservation of Volume When a solid object is reshaped (hammered, melted, recast), its total volume remains constant regardless of the new shape. $$ \text{Volume of Cube} = \text{Volume of Rectangular Sheet} $$ ### Step-by-Step Solution * Calculate the volume of the iron cube: $10^3 = 1000\text{ cm}^3$. * Let the dimensions of the rectangular sheet be $x$ and $5x$ based on the given $1:5$ ratio. The thickness (height) is $0.5\text{ cm}$. * Formulate the volume equation for the sheet: $\text{Length} \times \text{Breadth} \times \text{Thickness} = 5x \times x \times 0.5 = 2.5x^2$. * Equate the volumes: $2.5x^2 = 1000$. * Solve for $x^2$: $x^2 = \frac{1000}{2.5} = 400$. * Solve for $x$: $x = \sqrt{400} = 20\text{ cm}$. * Calculate the other side: $5x = 5(20) = 100\text{ cm}$. ### Exam Strategy & Shortcut Test the options directly into the volume formula. For option (a), $10 \times 50 \times 0.5 = 250$ (too small). For option (b), $20 \times 100 \times 0.5 = 1000$. This perfectly matches the cube's volume of $1000$, making (b) the immediate correct choice. ### Common Pitfall Misplacing the decimal when dividing by $0.5$ or $2.5$, leading to an incorrect root like $\sqrt{40}$ instead of $\sqrt{400}$. ### Final Answer Therefore, the correct answer is **20 cm, 100 cm**.